In mathematics, the Gibbs measure, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems.
The canonical ensemble gives the probability of the system X being in state x (equivalently, of the random variable X having value x) as
\[P(X=x)={\frac {1}{Z(\beta )}}\exp(-\beta E(x))\]where normalizing constant \(Z(\beta)\) is the partition function.
A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition:
- if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom.
The Hammersley–Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, Markov logic networks, and bounded rational potential games in game theory and economics.
A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density.
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